Finite Group Databasebeta

← #468193

Subgroups — #468193

Order 512.

1056 conjugacy classes of subgroups.

The subgroup lattice has 467 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2647C211yesmaximal, characteristic
2C2×#2066C212yesmaximal
3#14261C211yesmaximal, characteristic
4#11327C212yesmaximal
5#11323C211yesmaximal, characteristic
6#11268C211yesmaximal, characteristic
7#11228C212yesmaximal
8#11067C212yesmaximal
9#11052C211yesmaximal, characteristic
10#11050C211yesmaximal, characteristic
11#10833C211yesmaximal, characteristic
12C22×#106C2212yes—
13C22×C4⋊C8C2212yes—
14C22×#103C2212yes—
15C22×#103C2212yes—
16#2647—28no—
17C2×#400—22no—
18C2×#398—22no—
19C2×#391C2211yescharacteristic
20C2×#391C2211yescharacteristic
21C2×#391C2211yescharacteristic
22C2×#389C2212yes—
23C2×#389C2212yes—
24C2×#389C2211yescharacteristic
25C2×#384C2211yescharacteristic
26C2×#382C2211yescharacteristic
27C2×#376C2211yescharacteristic
28#2066—28no—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#336C2211yescharacteristic
35C2×#336C2211yescharacteristic
36C2×#336C2212yes—
37#1443—22no—
38#1442—22no—
39C4⋊#95C2211yescharacteristic
40C4⋊#95C2211yescharacteristic
41C23×Q8C2311yescharacteristic
42C23×C8C2312yes—
43C23×C8C2312yes—
44C22×C4⋊C4—22no—
45C22×C4⋊C4—22no—
46C22×C4⋊C4—22no—
47C22×C4⋊C4—21no—
48C22×C4⋊C4C2311yescharacteristic
49C22×C4⋊C4C2311yescharacteristic
50C22×C4⋊C4C2312yes—
51C22×C4⋊C4C2312yes—
52C22×C4⋊C4C2311yescharacteristic
53C22×C42C2311yescharacteristic
54C22×C42C2311yescharacteristic
55C22×C42C2311yescharacteristic
56C2×#106—24no—
57C2×#106—42no—
58C2×#106—42no—
59C2×C4⋊C8—24no—
60C2×C4⋊C8—24no—
61C2×C4⋊C8—42no—
62C2×#103—24no—
63C2×#103—24no—
64C2×#103—24no—
65C2×#103—42no—
66C2×#103—42no—
67C2×#103—42no—
68#400—216no—
69#398—216no—
70#391—22no—
71#391—24no—
72#391—42no—
73#391Q814yes—
74#389—24no—
75#389—24no—
76#389—24no—
77#389—42no—
78#389D418yes—
79#384—24no—
80#382—24no—
81#376—24no—
82C2×#95—22no—
83C2×#95—22no—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95—21no—
87C2×#95—22no—
88C2×#95—21no—
89C2×#95—21no—
90C2×#95—22no—
91C2×#95—22no—
92C2×#95D412yes—
93C2×#95D412yes—
94#340—24no—
95#340—24no—
96#340—42no—
97#340—42no—
98#340—42no—
99#336—22no—
100#336—22no—
101#336—42no—
102C22×Q8—21no—
103C22×Q8—22no—
104C22×Q8—22no—
105C22×Q8—22no—
106C22×Q8—21no—
107C22×Q8—21no—
108C22×Q8—22no—
109C22×Q8—21no—
110C22×Q8—21no—
111C22×Q8C2×Q812yes—
112C23×C4—22no—
113C23×C4—22no—
114C23×C4—21no—
115C23×C4—21no—
116C23×C4—21no—
117C23×C4—21no—
118C23×C4C2×D411yescharacteristic
119C23×C4C2×D411yescharacteristic
120C23×C4C2×D412yes—
121C23×C4C2×D412yes—
122C23×C4C2411yescharacteristic
123C22×C8—22no—
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8#4114yes—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—28no—
159C2×C4⋊C4—28no—
160C2×C4⋊C4—28no—
161C2×C4⋊C4—42no—
162C2×C4⋊C4—82no—
163C2×C4⋊C4—82no—
164C2×C4⋊C4—82no—
165C2×C4⋊C4—82no—
166C2×C4⋊C4—82no—
167C2×C4⋊C4—82no—
168C2×C4⋊C4—82no—
169C2×C4⋊C4C2×D414yes—
170C2×C4⋊C4C2×D414yes—
171C2×C4⋊C4C2×Q812yes—
172C2×C42—22no—
173C2×C42—22no—
174C2×C42—21no—
175C2×C42—21no—
176C2×C42—22no—
177C2×C42—22no—
178C2×C42—21no—
179C2×C42—22no—
180C2×C42—21no—
181C2×C42C2×D414yes—
182C2×C42C2×Q812yes—
183C2×C42#4112yes—
184#106—48no—
185C4⋊C8—48no—
186#103—48no—
187#103—48no—
188#95—28no—
189#95—28no—
190#95—28no—
191#95—24no—
192#95—24no—
193#95—28no—
194#95—42no—
195#95—42no—
196#95—42no—
197#95—44no—
198#95—82no—
199#95—82no—
200C24C22×D411yescharacteristic
201C2×Q8—22no—
202C2×Q8—22no—
203C2×Q8—24no—
204C2×Q8—24no—
205C2×Q8—41no—
206C2×Q8—41no—
207C2×Q8—42no—
208C2×Q8—42no—
209C2×Q8—42no—
210C2×Q8—42no—
211C2×Q8—42no—
212C2×Q8—42no—
213C2×Q8—42no—
214C2×Q8—42no—
215C2×Q8—42no—
216C2×Q8—42no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—21no—
230C22×C4—24no—
231C22×C4—21no—
232C22×C4—24no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—42no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—42no—
250C22×C4—41no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4#12014yes—
276C22×C4C22⋊Q814yes—
277C22×C4C22×D412yes—
278C22×C4C22×Q811yescharacteristic
279C22×C4C2×#4112yes—
280C22×C4C2×#4111yescharacteristic
281C22×C4H(5,𝔽2)11yescharacteristic
282C22×C4H(5,𝔽2)12yes—
283C22×C4H(5,𝔽2)11yescharacteristic
284C22×C42−1+411yescharacteristic
285C22×C42−1+412yes—
286C22×C42−1+411yescharacteristic
287C2×C8—24no—
288C2×C8—24no—
289C2×C8—24no—
290C2×C8—42no—
291C2×C8—42no—
292C2×C8—42no—
293C2×C8—42no—
294C2×C8—42no—
295C2×C8—42no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—42no—
300C2×C8—42no—
301C4⋊C4—24no—
302C4⋊C4—24no—
303C4⋊C4—24no—
304C4⋊C4—24no—
305C4⋊C4—24no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—88no—
316C4⋊C4—88no—
317C4⋊C4—88no—
318C4⋊C4—84no—
319C42—24no—
320C42—24no—
321C42—24no—
322C42—24no—
323C42—42no—
324C42—42no—
325C23C2×#12011yescharacteristic
326C23C2×C22⋊Q811yescharacteristic
327C23#53211yescharacteristic
328C23#53511yescharacteristic
329C23#53611yescharacteristic
330C23#53711yescharacteristic
331C23#53711yescharacteristic
332C23#57712yes—
333C23#57712yes—
334C23#57712yes—
335C23#57812yes—
336Q8—44no—
337Q8—44no—
338Q8—82no—
339Q8—82no—
340C2×C4—24no—
341C2×C4—24no—
342C2×C4—24no—
343C2×C4—24no—
344C2×C4—24no—
345C2×C4—24no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—44no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—44no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—44no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—44no—
371C2×C4—44no—
372C2×C4—44no—
373C2×C4—42no—
374C2×C4—44no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—82no—
378C2×C4—82no—
379C2×C4—81no—
380C2×C4—82no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4—82no—
384C2×C4—82no—
385C2×C4—81no—
386C2×C4—82no—
387C2×C4—82no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4—81no—
396C2×C4—82no—
397C2×C4—82no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4#53512yes—
401C2×C4#53712yes—
402C2×C4#53912yes—
403C2×C4#54011yescharacteristic
404C2×C4#54112yes—
405C2×C4#54111yescharacteristic
406C2×C4#54411yescharacteristic
407C2×C4#54411yescharacteristic
408C2×C4#54612yes—
409C2×C4#54712yes—
410C2×C4#54812yes—
411C2×C4D4×Q812yes—
412C2×C4#55312yes—
413C2×C4#55412yes—
414C2×C4#56212yes—
415C2×C4#56312yes—
416C8—44no—
417C8—82no—
418C22#240911yescharacteristic
419C22#243011yescharacteristic
420C22#256611yescharacteristic
421C22#261511yescharacteristic
422C22#262911yescharacteristic
423C22#266411yescharacteristic
424C22#282011yescharacteristic
425C22#300912yes—
426C22#301312yes—
427C22#308212yes—
428C22#308412yes—
429C22#317312yes—
430C22#317812yes—
431C22#322111yescharacteristic
432C22#322111yescharacteristic
433C22#322211yescharacteristic
434C22#322311yescharacteristic
435C22#324212yes—
436C22#324412yes—
437C22#325812yes—
438C22#325812yes—
439C22#326012yes—
440C22#326312yes—
441C4—44no—
442C4—44no—
443C4—82no—
444C4—84no—
445C4—82no—
446C4—81no—
447C4—84no—
448C4—81no—
449C4—161no—
450C4—161no—
451C4#270412yes—
452C4#271112yes—
453C4#272012yes—
454C4#272512yes—
455C2#1660411yescharacteristic
456C2#2431012yes—
457C2#2550912yes—
458C2#2919911yescharacteristic
459C2#2920511yescharacteristic
460C2#2967311yescharacteristic
461C2#2967611yescharacteristic
462C2#3000412yes—
463C2#3039412yes—
464C2#3283011yescharacteristic
465C2#3283211yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×#376 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#26472
C2×#20662
#142612
#113272
#113232
#112682
#112282
#110672
#110522
#110502
#108332